具單調與平滑收斂特性的二項樹選擇權定價模型分析

Binomial Option Pricing Models with Monotonic and Smooth Convergence Property

近年來，文獻上發現最具有效率性的二項樹選擇權定價模型，通常是當切割期數增加時，二項樹選擇權價格具有單調及平滑收斂到真值的特性，因為在這種情況底下，我們可以使用外插法來改進精確度。在本文中我們首先比較四種具有平滑收斂特性的二項樹模型在選擇權定價效率上的優劣，這些模型包括binomial Black-Scholes （BBS）模型、有彈性的二項樹模型（FB）、報酬函數平滑的（SPF）模型和一般化的CRR （GCRR）模型。其次，我們比較上述四種模型在計算delta及gamma兩個避險參數的效率性，數值分析的結果發現，上述四個模型在計算delta及gamma時也能夠產生單調及平滑收斂的特性。最後，GCRR-XPC模型是所有模型中，計算選擇權價格、delta及gamma最有效率的方法。

The recent literature indicates that the most efficient binomial models generally yield binomial option prices with monotonic and smooth convergence because one can apply the extrapolation formula to enhance the accuracy. In this paper, we first compare the pricing efficiency of four binomial models with monotonic and smooth convergence. These models include the binomial Black-Scholes (BBS) model of Broadie and Detemple (1996), the flexible binomial model (FB) of Tian (1999), the smoothed payoff (SPF) approach of Heston and Zhou (2000), and the generalized Cox-Ross-Rubinstein (GCRR) model of Chung and Shih (2007). Although these models have been proved to be efficient methods for pricing options, their efficiency for the calculation of delta and gamma is not known. To fill the gap of the literature, we then investigate the efficiency of these binomial models for calculating delta and gamma. The numerical results indicate that these models can also generate monotonic and smooth convergence estimates for deltas and gammas. Moreover, the GCRR-XPC model is the most efficient method to compute prices, deltas, and gammas for options.